1. Field
Embodiments of the present invention relate to geolocation of a non-cooperative target. More particularly, embodiments of the present invention relate to a system and method for using a globally convergent algorithm to determine the geolocation of a non-cooperative target.
2. Related Art
In order to locate a stationary ground target, sensor measurements and collection platform navigation data are processed either by a batch processing algorithm or a recursive estimator such as a Kalman filter. Such algorithms compute a target location estimate as the solution to a minimum variance optimization problem, with the answer being the point which best fits the sensor and navigation data in the weighted least-squares sense.
Nearly all geo-location algorithms, such as that used by a recursive estimator called the Extended Kalman Filter, require an initial target location estimate. However, when a limited number of sensor measurements are available, such as when using only a time difference of arrival (TDOA) and frequency difference of arrival (FDOA) of a signal received by two sensors, determining an initial location estimate can be very difficult due to the geometry of the measurements. For example, the isogram associated with a TDOA measurement is a branch of a hyperbola, and the isogram associated with an FDOA measurement is the locus of roots of a 4th order polynomial. For such measurements, the initial location estimate is often computed analytically as the intersection of the planar approximation of two measurement isograms. This is a difficult problem to solve, and if solved, can produce zero, one, or multiple intersections because of sensor errors and platform-target geometry.
Newton's Method is a nonlinear optimization algorithm, but is reliable only when an objective function being minimized is convex. In general, a function is convex at a point if and only if its Hessian matrix is positive semidefinite at that point. A lack of convexity at an initial location estimate can result in Newton's Method moving the location estimate away from the solution instead of towards it. Although the objective function being minimized for geo-location is typically convex near the solution, it may not be convex at distant points. The degree of convexity of the objective function is determined by the type of sensor measurements, number of measurements, and platform-target geometry. For example, when two collection platforms with poor geometry are using one TDOA and one FDOA measurement to locate a target, the objective function may possess a “saddle point”, i.e., the objective function will increase in some directions and decrease in others. The presence of a saddle point can result in divergence by Newton's Method unless the initial location estimate is close to the solution.
The Iterated Least-Squares algorithm is a simplification of Newton's Method in that the Hessian matrix is replaced with an approximation that is always at least positive semidefinite, if not positive definite. Although this guarantees that at each update the location estimate moves along a direction that initially does not increase the objective function, poor initialization can produce a huge change in the location estimate that overshoots the solution and results in convergence to a distant location.